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We study an abstract linear operator equation on a Banach space by using theinverse of the sum of two sectorial operators. We prove that the boundedness ofa special type of operator valued $H^\infty$-calculus is sufficient for maximalregularity of the solution. We apply the result to the abstract parabolicproblem, to give a maximal $L^{p}$-regularity condition. We also study theabstract hyperbolic problem and give a sufficient condition for the existenceof solution.

On the inverse of the sum of two sectorial operators